Normal Forms for Non-uniform Contractions

Let f be a measure-preserving transformation of a Lebesgue space (X,μ) and let F be its extension to a bundle E = X × R by smooth fiber maps Fx : Ex → Efx so that the derivative of F at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes Hx on Ex for μ-a.e. x so that the maps Px = Hfx ◦Fx ◦H−1 x are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such Hx and Px are unique up to a sub-resonance polynomial. As a consequence, we obtain the centralizer theorem that the coordinate change H also conjugates any commuting extension to a polynomial extension of the same type. We apply our results to a measurepreserving diffeomorphism f with a non-uniformly contracting invariant foliation W . We construct a measurable system of smooth coordinate changes Hx : Wx → TxW such that the maps Hfx ◦ f ◦ H−1 x are polynomials of sub-resonance type. Moreover, we show that for almost every leaf the coordinate changes exist at each point on the leaf and give a coherent atlas with transition maps in a finite dimensional Lie group.